Mathematical PhysicsStaff : J. Drummond, L. Frappat, L. Gallot, E. Ragoucy, E. Sokatchev and F. Thuillier Active retired staff : G. Clément, P. Sorba and R. Stora ANR: DIADEMS ; StrongInt Field theory and continuous symmetries are two cornerstones of modern theoretical particle physics. These two domains were complementarily used since a long time, but their merging really appeared in the 80's, with twodimensionnal conformal field theories, used in string theories. String theory is the most serious candidate for a unified theory of particle physics, including gravity. String/field theories and study of symmetries constitute the core of the subjects developped in the phymath team. An important place is given to infinite dimensional algebras (and / or deformed ones), appearing as mathematical properties (symmetry) of physical systems. In particular, integrable systems are central in the studies of the phymath group. They are among the main topics of theoretical physics that have experienced fundamental developments in the last twenty years. Integrability is imposed in quantum field theory for the development of nonperturbative methods for the calculation of physical quantities. This concept applies to any system, classical or quantum, possessing enough conservation laws for all physically relevant quantities to be calculated exactly. These conservation laws are associated with symmetries, but these symmetries can be hidden and their determination is a challenge for mathematics. The phymath group is a pole internationally recognized in the field of integrable systems. He produced many works on integrable systems in dimension 2, as the study of statistical mechanical models and / or condensed matter, especially spin chains models, the general treatment in field theory of diffusion in the presence of a factorized impurity, or of theories defined on onedimensional graphs. Relations with string theories are also strongly considered (see below). Recently, there have been remarkable developments in string theories, such as Witten's twistor approach, BernKosowerDixon method for perturbative calculation to large orders, or AldayMaldacena proposal for the description dual scattering amplitudes. They have led to great progress in the calculation of scattering amplitudes, avoiding the traditional calculation of Feynman diagrams. Most of these results took place in maximally supersymmetric YangMills theories (N = 4 SYM). These theories are very similar to QCD, with whom they share many important properties. The group is a leader in the calculation of scattering amplitudes of these theories. More formal studies of field theories and functional integrals are also conducted within the phymath group. They relate in particular nonAbelian ChernSimons theories and DeligneBeilinson cohomology. In addition, the phymath group leads numerous studies on exact solutions in general relativity: for example, new black hole solutions and black membranes were obtained as part of dilatonic theories of gravity inspired by string theory. To sumarize, the main research areas studied by the group are:

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